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Please,please help me The length of a focal chord of the parabola y2= 4ax at a distance b from the vertex is C. Then

The length of a focal chord of the parabola y2 = 4ax at a distance b from the vertex is C. Then

  • Option 1)

    2a2 = bc 

  • Option 2)

    a3 = b2c

  • Option 3)

    ac = b2

  • Option 4)

    b2c = 4a3

 
Answers (1)
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Find intersections A(x1, y1), B(x2, y2) of P & L

Eliminate y from (1) & (2):

m2x2 - 2am x + m2 a2 = 4ax 

m2 x2-2a(m + 2) x + m2 a2 = 0                     ..............(3)

x1, x2 are the roots, x1 + x2 = 2a (m+2)/m2 ;; x1 x2 = a2         .........(4)

Eliminate x from (1) & (2):

y = m(y2/4a) - ma

my2 - 4a y-4a2 m = 0                        ..........(5)

y1, y2 are the roots, y1 + y2 = 4a/m     ;; y1 y2 = - 4a2            ........(6)

Now length of focal chord = c = AB

c2 = (x1 - x2)2 + (y1 -y2)2

    = (x1 + x2)2 - 4x1 x2 + (y1 + y2)2 - 4y1 y2

    =4a2 (m4 + 4 + 4m2)/m4 - 4a2 + 16a2/m2+16a2

    = 16a2 (m2 +1)2/m4

    = 16 a2 (a2 / b2)2            .............using (3)

    \Rightarrow    b2c = 4a3

    Modulus as "a" can be +ve or -ve.

 


Option 1)

2a2 = bc 

Incorrect

Option 2)

a3 = b2c

Incorrect

Option 3)

ac = b2

Incorrect

Option 4)

b2c = 4a3

Correct

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